Today is the second day of Rosh Hashanah and the synagogue reading is Genesis 22. Why did the rabbis make this designation?

The surface connection is easily seen. Isaac was spared when God provided a ram in the thicket. On the Feast of Trumpets we sound the shofar. That's obvious, but lets go deeper.

The shofar makes a clear and penetrating sound even though the ram died long before. Every ram starts out as a little lamb. This is the designation bestowed by John the Baptist upon Yeshua (John 1:29). Yeshua died long ago, but His voice is clearly heard to today by those who listen. (John 10:27)

Every new year marks a season of new beginnings. The Lord is speaking. Isaiah tells us to be listening.

Isaiah 30:21 - Your ears will hear a word behind you saying: "This is the way, walk in it. When you turn to the right, or when you turn to the left."

Shema Yisrael (Deuteronomy 6:4) was spoken to Israel, but to you who are grafted in, you are Abraham's seed. You also must hear and obey.